Relevant For...

Why some people say it's prime: Its positive divisors are 1 and itself.

Why some people say it's not prime: It doesn't have exactly two positive divisors.

The statement "1 is prime" is \( \color{red}{\textbf{false}}\).

Proof: The definition of a prime number is a positive integer that has exactly two positive divisors. However, 1 only has one positive divisor (1 itself), so it is not prime.

Rebuttal: That's not the definition of a prime number! A prime number is a positive integer whose positive divisors are exactly 1 and itself.

Reply: This might be part of the definition of a prime number that you've seen elsewhere, if another clause of the definition specified that "all prime numbers are \(">1"\) or \( "\geq 2."\) However, any correct definition will specifically exclude 1. (If you want to stick with that definition, it is now "a prime number is a positive integer greater than 1 whose positive divisors are exactly 1 and itself.")

Rebuttal: Why shouldn't 1 be prime?

Reply: This is just a matter of definition. Mathematicians love to define things; they decide that 1 shouldn't be prime, because they can do so. Of course, mathematicians also have reasons when defining things, and are not just making this decision at whim. In this case, one reason is the fundamental theorem of arithmetic:

Every positive integer greater than 1 can be represented uniquely (up to the order) as the product of one or more prime numbers.

If 1 is a prime number, this theorem would break down, since \(6 = 2 \times 3 = 1 \times 2 \times 3 = 1 \times 1 \times 2 \times 3 = \ldots\), making it not unique. Of course, we can change how the fundamental theorem of arithmetic is stated:

Every positive integer greater than 1 can be represented uniquely (up to the order) as the product of one or more prime numbers that are not 1.

But now we've pushed the complication of 1 being a "special case" from the definition to an important theorem. If this is true for most theorems involving the set of "primes + 1," then many will claim that, in fact, it would be more elegant for the set of primes to not include 1, thereby simplifying the statement of all of the theorems. It's a human choice where to put the cutoff for every definition. However, since mathematicians are trying to create as much elegance and simplicity as possible, this motivated the decision to make the definition of primes such that 1 is not a prime.

Want to make sure you've got this concept down? Try these problems:

Can a composite number, times a prime number, divided by another prime number ever equal a prime number?